A Critical Parabolic Sobolev Embedding via Littlewood-Paley Decomposition

Transcription

1 International Mathematical Forum, Vol. 7, 01, no. 35, A Critical Parabolic Sobolev Embedding via Littlewood-Paley Decomposition H. Ibrahim Lebanese University, Faculty of Sciences Mathematics Department, Hadeth, Beirut, Lebanon & School of Arts and Sciences, Mathematics Department Lebanese International University LIU) Beirut Campus, Al-Mouseitbeh P.B. Box , Beirut, Lebanon hassan.ibrahim@liu.edu.lb Abstract In this paper, we show a parabolic version of the Ogawa type inequality in Sobolev spaces. Our inequality provides an estimate of the L norm of a function in terms of its parabolic BMO norm, with the aid of the square root of the logarithmic dependency of a higher order Sobolev norm. The proof is mainly based on the Littlewood-Paley decomposition and a characterization of parabolic BMO spaces. Mathematics Subject Classifications: 4B35, 54C35, 4B5, 39B05 Keywords: Littlewood-Paley decomposition, logarithmic Sobolev inequalities, parabolic BMO spaces, Lizorkin-Triebel spaces, Besov spaces 1 Introduction and main results In order to study the long-time existence of a certain class of singular parabolic problems, Ibrahim and Monneau [9] made use of a parabolic logarithmic Sobolev inequality. They proved that for f W m,m R n+1 ), m, n N and m > n+, the following estimate takes place with log + x = maxlog x, 0)): L R n+1 ) C1 + BMO a R n+1 )1 + log + W m,m R n+1 ))), 1.1.1)

2 1706 H. Ibrahim for some constant C = Cm, n) > 0. Here BMO a stands for the anisotropic Bounded Mean Oscillation space with the parabolic anisotropy a =1,...,1, ) R n+1 see Definition.1), while W m,m stands for the parabolic Sobolev space see Definition.). The above estimate, after also being proved on a bounded domain Ω T =0, 1) n 0,T) R n+1, 1.1.) was successfully applied in order to obtain some a priori bounds on the gradient of the solution of particular parabolic equations leading eventually to the long-time existence see [9, Proposition 3.7] or [8, Theorem 1.3]). The bounded version of 1.1.1) see [9, Theorem 1.]) reads: if f W m,m Ω T ) with m > n+, then: L Ω T ) C1 + a BMO ΩT ) 1 + log+ W m,m Ω T ))), 1.1.3) where C = Cm, n, T ) > 0 is a positive constant, and BMO a ΩT ) = BMO a Ω T ) + L 1 Ω T ) ) Indeed, the fact that inequality 1.1.1) does not hold on Ω T with a positive constant C = C m, n, T ) can be easily understood by applying this inequality to the function f =C +ɛ) W m,m Ω T ) with ɛ>0. In this case L Ω T ) = C + ɛ, BMO a Ω T ) = 0, and hence a contradiction. However, working on R n+1, the same function f could not be used since f /W m,m R n+1 ). Let us indicate that both inequalities 1.1.1) and 1.1.3) still hold for vector-valued functions f =f 1,...,f n,f n+1 ) W m,m R n+1 )) n+1 with m > n+ and the natural change in norm. The elliptic version of 1.1.1) was showed by Kozono and Taniuchi in [1]. Indeed, they have showed that for f W s p Rn ), 1 <p<, the following estimate holds: L R n ) C1 + BMOR n )1 + log + W s p R n ))), sp > n, 1.1.5) for some C = Cn, p, s) > 0. Here BMO is the usual elliptic/isotropic bounded mean oscillation space defined via Euclidean balls). The main advantage of 1.1.5) is that it was successfully applied in order to extend the blow-up criterion of solutions to the Euler equations originally given by Beale, Kato and Majda in [1]. This blow-up criterion was then refined by Kozono, Ogawa and Taniuchi [11], and by Ogawa [13], showing weaker regularity criterion.

3 Critical parabolic Sobolev embedding 1707 The proof of inequality 1.1.1) is based on the analysis in anisotropic Lizorkin-Triebel, Besov, Sobolev and BMO a spaces. This is made via Littlewood- Paley decomposition and various Sobolev embeddings. In fact, some of the technical arguments were inspired by Ogawa [13] in his proof of the sharp version of 1.1.5) that reads: if g L R n ) and f := g Wq 1 R n ) L R n ) for n<q, then there exists a constant C = Cq) > 0 such that: ) ) 1/ L R n ) Cq) 1+ BMOR n ) log + W 1 q R n ) + g L R n )) ) It is worth mentioning that the original type of the logarithmic Sobolev inequalities 1.1.5) and 1.1.6) was found in Brézis and Gallouët [3]. The Brézis- Gallouët-Wainger inequality states that the L norm of a function can be estimated by the Wp n/p norm with the partial aid of the Wr s norm with s > n/r and 1 r. Precisely, ) p 1 p L R n ) C 1 + log1 + W s r R n ))) 1.1.7) holds for all f Wp n/p R n ) Wr srn ) with the normalization n/p W p R =1. n ) Originally, Brézis and Gallouët [3] obtained 1.1.7) for the case n = p = r = s =, where they applied their inequality in order to prove global existence of solutions to the nonlinear Schrödinger equation. Later on, Brézis and Wainger [4] obtained 1.1.7) for the general case, and remarked that the power p 1 in p 1.1.7) is optimal in the sense that one can not replace it by any smaller power. However, it seems that little is known about the sharp constant in 1.1.7). Coming back to inequalities 1.1.1), 1.1.5) and 1.1.6), the natural question that arises is the following: why does the inequality 1.1.1) seems to be the parabolic extension of 1.1.5) although the proof is inspired as mentioned above) from that of 1.1.6) given by Ogawa [13]? The answer to this question is partially contained in [9, Remark.14] where the authors pointed out that the well-known relation between elliptic/isotropic Lizorkin-Triebel and BMO spaces see [13, Proposition.3]) will not be used in the proof of 1.1.1) even though it seems to be valid without giving a proof) in the parabolic/anisotropic framework. The relation is the following: F 0,a, BMO a, 1.1.8) is the homogeneous parabolic Lizorkin-Triebel space see Defini- where tion.3). F 0,a,

4 1708 H. Ibrahim In this paper, we show a parabolic version of the logarithmic Sobolev inequality 1.1.6) basically using the equivalence 1.1.8) that is shown to be true see Lemma 3.1). This answers the question raised above. Our study takes place on the whole space R n+1 and on the bounded domain Ω T. A comparison in some special cases) of our inequality with 1.1.1) is also discussed. Before stating our main results, we define some terminology. A generic element in R n+1 will be denoted by z =x, t) R n+1 where x =x 1,...,x n ) R n is the spatial variable, and t R is the time variable. For a given function g, the notation i g stands for the partial derivative with respect to the spatial variable: i g = xi g := g x i, i =1,..., n. In this case n+1 g = t g := g. We t also denote x s g, s N, any derivative with respect to x of order s. Moreover, we denote the space-time gradient by g := 1 g,..., n g, n+1 g). Finally, we denote X := max f 1 X,..., f n X, f n+1 X ) for any vector-valued function f =f 1,...,f n,f n+1 ) X n+1 where X is any Banach space. Throughout this paper and for the sake of simplicity, we will drop the superscript n +1 from X n+1. Following the above notations, our first theorem reads: Theorem 1.1 Parabolic Ogawa inequality on R n+1 ). Let m, n N with m > n+. Then there exists a constant C = Cm, n) > 0 such that for any function g L R n+1 ) with f =f 1,...,f n,f n+1 )= g W m,m R n+1 ),we have: ) ) 1/ L R n+1 ) C 1+ BMO a R n+1 ) log + W m,m R n+1 ) + g L R n+1 )) ) Remark 1. Inequalities 1.1.1) and 1.1.9) have the same order of the higher regular term. As a consequence, inequality 1.1.9) can also be applied in order to establish the long-time existence of solutions of the parabolic problems studied in [8, 9]. Our next theorem concerns a similar type inequality of 1.1.9), but with functions g and f defined over Ω T given by 1.1.)). Before stating this result, we first remark that in the case of functions f = g defined on a bounded domain, we formally have by Poincaré inequality): g L C L, where C>0is a constant depending on the measure of the domain. Moreover, since L C 1 C γ,γ/ C W m,m with C 1,C > 0,

5 Critical parabolic Sobolev embedding 1709 the above two estimates imply that the term g L should be dropped from inequality 1.1.9) when dealing with functions defined over bounded domains. Indeed, we have: Theorem 1.3 Parabolic Ogawa inequality on a bounded domain). Let f W m,m Ω T ) with m > n+. Then there exists a constant C = Cm, n, T ) > 0 such that: ) ) 1/ L Ω T ) C 1+ a BMO ΩT ) log + W m,m Ω T ), ) where the norm BMO a ΩT ) is given by 1.1.4). Remark 1.4 Inequality ) is sharper than 1.1.3) by the simple observation that x 1/ 1+x for x 0. In other words, inequality ) implies 1.1.3) with the same positive constant C = Cm, n). In the same spirit of Remark 1.4, our last theorem gives a comparison between inequality 1.1.1) and 1.1.9) for a certain class of functions g, and for particular space dimensions. Theorem 1.5 Comparison between parabolic logarithmic inequalities). Let n = 1,, 3 and m N satisfying m > n+. There exists a constant C = Cm, n) > 0 such that for the class of functions g L R n+1 ) with g L R n+1 ) 1, and f = g W m,m R n+1 ), we have: log + W m,m R n+1 ) + g L R n+1 )) ) 1/ C1 + log + W m,m R n+1 ) ), ) and hence inequality 1.1.9) implies 1.1.1) for possibly a different positive constant C. 1.1 Organization of the paper This paper is organized as follows. In Section, we present some definitions and the main tools used in our analysis. This includes parabolic Littlewood- Paley decomposition and various Sobolev embeddings. Section 3 is devoted to the proof of Theorem 1.1 estimate on the entire space R n+1 ) using mainly the equivalence 1.1.8) that we also show in Lemma 3.1. In Section 4, we give the proof of Theorem 1.3 estimate on the bounded domain Ω T ). Finally, in Section 5, we give the proof of Theorem 1.5.

6 1710 H. Ibrahim Preliminaries and basic tools In this section, we define the fundamental function spaces used in this paper. We also recall some important embeddings..1 Parabolic BMO a and Sobolev spaces Each coordinate x i, i =1,..., n is given the weight 1, while the time coordinate t is given the weight. The vector a =a 1,...,a n,a n+1 )=1,...,1, ) R n+1 is called the n + 1)-dimensional parabolic anisotropy. For this given a, the action of μ [0, ) onz =x, t) is given by μ a z =μx 1,...,μx n,μ t). For μ>0 and s R we set μ sa z =μ s ) a z. In particular, μ a z =μ 1 ) a z and ja z = j ) a z, j Z. For z R n+1, z 0, let z a be the unique positive number μ such that: x 1 μ + + x n μ + t μ 4 =1 and let z a = 0 for z = 0. The map a is called the parabolic distance function which is C. In the case where a =1,...,1) R n+1, we get the usual Euclidean distance z =x x n + t ) 1/. Denoting O R n+1, any open subset of R n+1, we are ready to give the definition of the first two parabolic spaces used in our analysis. Definition.1 Parabolic bounded mean oscillation spaces). A function f L 1 loc O) defined up to an additive constant) is said to be of parabolic bounded mean oscillation, f BMO a O), if we have: BMO a O) = sup inf Q O c R ) 1 f c < +,..1) Q Q where Q denotes for z 0 Oand r>0) an arbitrary parabolic cube: Q = Q r z 0 )={z R n+1 ; z z 0 a <r}. Definition. Parabolic Sobolev spaces). Let m N. We define the parabolic Sobolev space W m,m O) as follows: W m,m O) ={f L O); r t s x f L O), r, s N such that r + s m}, with W m,m O) = m j=0 r+s=j r t s x f L O).

7 Critical parabolic Sobolev embedding Parabolic Lizorkin-Triebel and Besov spaces Along with the above parabolic distance a, the Littlewood-Paley decomposition is now recalled. Let θ C 0 Rn+1 ) be any cut-off function satisfying: θz) =1 if z a 1 θz) =0 if z a...) Let ψz) =θz) θ a z). We now construct a smooth compactly supported) parabolic dyadic partition of unity ψ j ) j Z by letting ψ j z) =ψ ja z), j Z,..3) satisfying j Z ψ jz) = 1 for z 0. Define ϕ j, j Z, as the inverse Fourier transform of ψ j, i.e. ˆϕ j = ψ j where we let It is worth noticing that ϕ j satisfies: ϕ := ϕ 0...4) ϕ j z) = n+)j ϕ ja z), j Z and z R n+1...5) The above Littlewood-Paley decomposition asserts that any tempered distribution f S R n+1 ) can be decomposed as: f = j Z ϕ j f with the convergence in S /P modulo polynomials). Here SR n+1 ) is the usual Schwartz class of rapidly decreasing functions and S R n+1 ) is its corresponding dual, represents the space of tempered distributions. We now define parabolic Lizorkin-Triebel spaces. Definition.3 Parabolic homogeneous Lizorkin-Triebel spaces). Given a smoothness parameter s R, an integrability exponent 1 p<, and a summability exponent 1 q 1 q< if p = ). Let ϕ j be given by..5), we define the parabolic homogeneous Lizorkin-Triebel space F p,q s,a as the space of all functions f S R n+1 ) with finite quasi-norms ) 1/q F p,q s,a R n+1 ) = sqj ϕ j f q <, j Z L p R n+1 ) and the natural modification for q =.

8 171 H. Ibrahim As a convention, for s R, and 1 q<, we denote f + F,qR s,a n+1 ) = sqj ϕ j f q ) 1/q L R n+1 ) and j 1 f F,qR s,a n+1 ) = j 1..6) sqj ϕ j f q ) 1/q L R n+1 )...7) The space F 0,a p, can be identified with the parabolic Hardy space H p,a R n+1 ), 1 p<, having the following square function characterization stated informally as: { H p,a R n+1 )= f S R n+1 ); ϕ j f ) 1/ L }. p..8) j Z This identification between the above two spaces is the following: Theorem.4 Identification between H p,a and F 0,a p, ). See Bownik [].) For all 1 p<, we have F 0,a p, Rn+1 ) H p,a R n+1 ). Another useful space throughout our analysis is the parabolic inhomogeneous Besov space. The main difference in defining this space is the choice of the parabolic dyadic partition of unity that is now altered. Indeed, we take ψ j ) j 0 satisfying: ψ j := ψ j defined by..3) if j 1 ψ j := θ defined by..) if j =0...9) Again, it is clear that j 0 ψ jz) = 1, but now for all z R n+1, and in exactly the same way as above, we can rewrite the Littlewood-Paley decomposition with ˆϕ j = ψ j,j 0, ψ j is given by..9)...10) We then arrive to the following definition: Definition.5 Parabolic inhomogeneous Besov spaces). Given a smoothness parameter s R, an integrability exponent 1 p, and a summability exponent 1 q, we define the parabolic inhomogeneous Besov space Bp,q s,a as the space of all functions u S R n+1 ) with finite quasi-norms u B s,a p,q = j 0 sqj ϕ j u q L p R n+1 )) 1/q <, ϕ j is given by..10)

9 Critical parabolic Sobolev embedding 1713 and the natural modification for q =, i.e. u B s,a = sup sj ϕ p, j u L p R n+1 ), ϕ j is given by..10)...11) j 0 For a detailed study of anisotropic Lizorkin-Triebel and Besov spaces, we refer the reader to Triebel [16]..3 Embeddings of parabolic Besov and Sobolev spaces We present two embedding results from Johnsen and Sickel [10], and Stöckert [14]. Theorem.6 Embeddings of Besov spaces).see Johnsen and Sickel [10].) Let s, t R, s>t, and 1 p, r satisfy: s n+ = t n+. Then for any p r 1 q we have the following continuous embedding: B s,a p,q Rn+1 ) B t,a r,q Rn+1 )...1) Proposition.7 Sobolev embeddings in Besov spaces).see Stöckert [14].) Let m N, then we have: W m,m R n+1 ) B m,a, Rn+1 )...13) 3 Proof of Theorem 1.1 In this section we give the proof of Theorem 1.1. We start by showing the equivalence 1.1.8) whose isotropic version can be found in Triebel [15], and Frazier and Jawerth [6]. Lemma 3.1 Equivalence between F 0,a, and BMOa ). We have F 0,a, Rn+1 ) BMO a R n+1 ). Precisely, there exists a constant C>0 such that: C 1 F 0,a BMO a CḞ 0,a ),, Proof. See [7, Lemma 3., page 6]. A basic estimate is now shown in the following lemma. Lemma 3. Logarithmic estimate in parabolic Lizorkin-Triebel spaces). Let γ>0be a positive real number. Then, for f F 0,a,1 with f + F γ,a, Rn+1 ) and

10 1714 H. Ibrahim f F γ,a that:, Rn+1 ) F 0,a C,1 are both finite, there exists a constant C = Cn, γ) > 0 such 1+ F 0,a, log + f + F γ,a + f, F γ,a, ) ) 1/ ). 3.3.) Proof. We first indicate that the constant C = Cn, γ) > 0 may vary from line to line in the proof which is divided into two steps. Step 1 First estimate on F 0,a ). Let N N. We compute,1 F 0,a,1 j< N γj γj ϕ j f + L j< N j N ϕ j f + γj γj ϕ j f L j>n j N L C γ γn ) 1/ γj ϕ j f L +N +1) 1/ ) 1/ ϕ j f L +C γ γn ) 1/ γj ϕ j f L j>n C γ γn f F γ,a + CN +1) 1/ Ḟ 0,a,, with C γ = 1 1/. 1) As a conclusion we may write γ F 0,a C N +1) 1/,1 F 0,a + γn f + F γ,a + f,, + C γ γn f + F γ,a,, F γ,a, Step Optimization in N). We optimize 3.3.3) in N by setting: ) ) ) N =1 if f + F γ,a + f, F γ,a, γ Ḟ 0,a., Then it is easy to check using 3.3.3)) that Ḟ 0,a C,1 F 1+ f + 0,a log + F γ,a + f,, F 0,a, In the case where f + F γ,a + f, that N = Nβ) = log + γ β F γ,a, f + F γ,a F γ,a, ) 1/ ) > γ F 0,a, we take 1 β< γ such, ), + f Ḟ 0,a, F γ,a, 1 N. In fact this is valid since the function Nβ) varies continuously from N1) to N γ )=1+N1) on the interval [1, γ ]. Using 3.3.3) with the above choice

11 Critical parabolic Sobolev embedding 1715 of N, we obtain: F 0,a,1 C 1/ log + β γ f + F γ,a + f, F 0,a, F γ,a, C f+ log + F γ,a + f, γ log ) 1/ F 0,a, where for the second line we have used the fact that f + log + β<log + F γ,a + f, F 0,a, )) 1/ F γ,a + γ/, F γ,a, F γ,a,. β )) 1/ F γ,a + γ/, The above computations again imply 3.3.4). By using the inequality: x log e + x)) y 1/ C1 + xloge + y)) 1/ ) for 0 <x 1, F γ,a, β Ḟ γ,a,, x log e + y )) 1/ Cxloge + y)) 1/ x in 3.3.4), we directly arrive to our result. for x>1, 3.3.5) We now present the proof of our first main result. Proof of Theorem 1.1. First let us mention that the constant C = Cm, n) > 0 appearing in the following proof may vary from line to line. We will show inequality 1.1.9) in the scalar-valued version, i.e. by considering f = f i = i g for some fixed i =1,...,n+ 1. The vector-valued version can then be easily deduced. The proof requires estimating all the terms of inequality 3.3.). We start with the obvious estimate see 3.3.1)): F 0,a C BMO a ), The remaining terms will be estimated in the following three steps. n+ Step 1 An upper bound on f + F γ,a ). Set η =m > 0. Choose γ such, that: 0 <γ<η.

12 1716 H. Ibrahim We compute see..6)): f + F γ,a, = ) 1/ γj ϕ j f j 1 L C sup ηj ϕ j f L 3.3.7) j 1 ) 1/ with C = j 1 γ η)j < +. Note that the sequence of functions ϕ j ) j 1 given in 3.3.7) can be identified with those given in..11). Hence we may write sup j 1 ηj ϕ j f L sup ηj ϕ j f L, ϕ j is given by..10), j 0 and then using 3.3.7)) we obtain: f + F γ,a, C B η,a, ) Using..1) with s =m, p =,q =, t = η and r =, we deduce that: Therefore, by..13), we get which, together with 3.3.8), give: B m,a, B η,a,. W m,m B m,a, B η,a, f + F γ,a C, W m,m ) Step An upper bound on f F γ,a ). In this step, we will use the fact, that i g = f i for which we keep denoting it by f, i.e. f = f i ) for some i =1,...,n+ 1, with g L R n+1 ). For z R n+1, define and Using..5) we obtain: Φz) = i ϕ)z), ϕ is given by..4), ) Φ j z) = n+)j Φ ja z) for all j ) i ϕ j )z) = j Φ j z) if i =1,...,n

13 Critical parabolic Sobolev embedding 1717 i ϕ j )z) = j Φ j z) if i = n ) We now compute see..7), ) and 3.3.1)): ) 1/ f F =, γj ϕ j f j 1 L ) C sup Φ j g L, ) j 1 where the constant C is given by: C = j1 γ) if i =1,...,n j 1 C = j γ) if i = n +1, j 1 which is finite 0 <C<+ under the choice 0 <γ<1. In order to terminate the proof, it suffices to show that Φ j g L C g L, which can be deduced, by translation and dilation invariance, from the following estimate: Φ g)0) C g L ) Indeed, define the positive radial decreasing function hr) = h z ) as follows: hr) = sup Φz). z r From ), we remark that the function Φ is the inverse Fourier transform of a compactly supported function. Hence, we have: and the asymptotic behavior h0) = Φ L < +, ) hr) C r n+ for all r )

14 1718 H. Ibrahim We compute taking Sr n as the n-dimensional sphere of radius r): Φ g)0) Φ z) gz) dz R n+1 ) Φ z) gz) dσz) dr Using ) and ) we deduce that: 0 r n hr)dr = 0 S n r ) C r n hr)dr g L ) r n hr)dr + 1 C h0)dr + 0 C Φ L +1) 1 1 r n hr)dr ) r n dr rn+ which, together with ), directly implies ). As a conclusion, we obtain see )): f F γ,a C g L ), Step 3 A lower bound on Ḟ 0,a and conclusion). Remarking that,1 L = ϕ j f L j Z F 0,a,1 when ˆf0) = 0, the estimates 3.3.), 3.3.6), 3.3.9) and ) lead directly to the proof. 4 Proof of Theorem 1.3 For the sake of simplicity, we only give the proof in the framework of one spatial dimensions x = x 1. The extension to the multi spatial dimensions can be easily deduced and will be made clear later in this section. Again, the constant C>0that will appear in the following proof may vary from line to line but will only depend on m and T Proof of Theorem 1.3. We first remark that the function f can be extended by continuity to the boundary Ω T of Ω T. Following the same notations of Ibrahim and Monneau [9], we take f as the extension of f over Ω T = 1, ) T,T)

15 Critical parabolic Sobolev embedding 1719 given by: fx, t) = fx, t) = with λ j = 1 j and m 1 j=0 m 1 j=0 fx, t) = c j f λ j x, t) for 1 <x<0, 0 t T 4.4.1) c j f1 + λ j 1 x),t) for 1 <x<, 0 t T, m 1 j=0 c j λ j ) k = 1 for k =0,...,m 1. For the extension with respect to the time variable t, we use the same extension 4.4.1) summing up only to m 1. The above extension 4.4.1) has been made in order to have see for instance Evans [5]) f W m,m Ω T ) and f W m,m e Ω T ) C W m,m Ω T ). 4.4.) Now let Z 1 Z be two subsets of Ω T defined by: Z 1 = {x, t); 1/4 <x<5/4 and T/4 <t<5t/4} and Z = {x, t); 3/4 <x<7/4 and 3T/4 <t<7t/4}. We take the cut-off function Ψ C 0 R ), 0 Ψ 1 satisfying: Ψx, t) = 1 for x, t) Z ) Ψx, t) = 0 for x, t) R \Z. From 4.4.), we easily deduce that Ψ f W m,m R ) and Ψ f W m,m R ) C W m,m Ω T ) ) Hence we can apply the scalar-valued version of inequality 1.1.9) see Remark??) with i = 1, i.e. 1 g = f; the new function for which we give the same notation) f =Ψ f W m,m R ) and g L R ) given by gx, t) = x 0 Ψy, t) fy, t)dy.

16 170 H. Ibrahim Since Ψ f is of compact support, and again by the extension 4.4.1)) f L eω T ) C W m,m Ω T ), we deduce that g L R ) C f L eω T ) C W m,m Ω T ) ) Collecting the above arguments namely 4.4.4) and 4.4.5)) together with the fact that see Ibrahim and Monneau [9]) Ψ f BMO a R ) C BMO a ΩT ), inequality 1.1.9) gives: L Ω T ) Ψ f L R ) C 1+ Ψ f BMO a R ) C log + Ψ f W m,m 1+ a BMO ΩT ) log + W m,m Ω T ) Notice that in the first line of the above inequalities we have used that Ψ = 1 in Ω T. Remark 4.1 The inequality 1.1.9) used in the previous proof could have also been used for i =. In this case we simply take gx, t) = t 0 Ψx, s) fx, s)ds. Remark 4. In the case of multi spatial dimensions x i, i =1,...,n, we simultaneously apply the extension 4.4.1) to each spatial coordinate while fixing all the other coordinates including time t. However, the extension with respect to t is kept unchanged. ) ) 1/ R ) + g L R )) ) ) 1/. 5 Comparison between parabolic logarithmic inequalities In this section we give the proof of Theorem 1.5. Throughout all this section, we only consider isotropic function spaces, i.e. a =1,...,1) R n+1. We only deal with the parabolic function space W m,m. As usual, the constant C = Cm, n) > 0 may differ from line to line. First of all, we remark that estimate ) turns out to be true using the trivial identity x 1/ 1+x) if A := g L C for W m,m 1, or if B := g L / W m,m C for W m,m 1. This will be proved in the forthcoming arguments. We start with the following lemmas:

17 Critical parabolic Sobolev embedding 171 Lemma 5.1 Let n =1,, 3, s = n+1, and m N satisfying m > n+. For any g L R n+1 ) with f = g W m,m R n+1 ), we have g Ḣs R n+1 ) and g Ḣs W m,m ) The norm in the homogeneous Sobolev space Ḣs is given by Ḣ s = R n+1 ξ s ˆfξ) dξ where is the usual Euclidean distance. Proof. Follows directly since 1 s m, using the definition of the norm in Ḣ s. Lemma 5. Under the same hypothesis of Theorem 1.5, we have: g L C 1+ W m,m log e + )) 1/ W m,m + g L. 5.5.) W m,m Proof. We consider the isotropic a =1,...,1)) homogeneous dyadic partition of unity ψ j ) j Z with j Z ψ j = 1 and ˆϕ j = ψ j. Fix some 0 <γ<1, and take an arbitrary N N. We write: g L j 0 ϕ j g L + N ϕ j g L + ϕ j g L ) j>n j=1 We estimate the right-hand side of 5.5.3). Benstein s inequality gives: We let s = n+1. Using 5.5.4), we compute: ϕ j g L C j 0 j 0 Again, using 5.5.4), we obtain: ϕ j g L C n+1 )j ϕ j g L ) sj ϕ j g L C 1 g s L ) N ϕ j g L C j=1 N sj ϕ j g L CN 1/ g Ḃs,, j=1 which, together with the fact that Ḃs, Ḣs, and estimate 5.5.1) of Lemma 5.1, yield: N j=1 ϕ j g L CN 1/ W m,m )

18 17 H. Ibrahim The last term of the right-hand side of 5.5.3) can be estimated as follows: ϕ j g L = ) jγ jγ ϕ j g L ) γn γ 1 γ g Ḃγ,. j>n j>n 5.5.7) We know that Ḃγ, Ċγ ; the homogeneous Hölder space whose semi-norm can be estimated as follows: gz 1 ) gz ) g Ċγ = sup z 1 z z 1 z γ W m,m + g L. This, together with 5.5.7) yield: j>n ϕ j g L C γn W m,m Combining 5.5.3), 5.5.5), 5.5.6) and 5.5.8), we finally get: g L C 1+N 1/ W m,m + γn W m,m + g L ) ) + g L ) ). By optimizing as in Step of Lemma 3.) in N the above inequality, the proof easily follows. We are now ready to give the proof of Theorem 1.5. Proof of Theorem 1.5. As it was already mentioned in the beginning of this section, the proof relies on considering two cases. Case 1 W m,m 1). Let A := g L. Using inequalities 3.3.5) and 5.5.), we obtain: A C[1 + loge +1+A)) 1/ ], which directly implies that A C, and hence ) is obtained. Case W m,m g L W m,m 1). Dividing inequality 5.5.) by W m,m )) g 1/ L C 1+ log e +1+ W m,m., we obtain: Letting B := g L / W m,m, we can easily see that B satisfies as the term A in Case 1): B C[1 + loge +1+B)) 1/ ], which shows that B C, and the proof is done.

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20 174 H. Ibrahim [13] T. Ogawa, Sharp Sobolev inequality of logarithmic type and the limiting regularity condition to the harmonic heat flow, SIAM J. Math. Anal., ), pp electronic). [14] B. Stöckert, Remarks on the interpolation of anisotropic spaces of Besov- Hardy-Sobolev type, Czechoslovak Math. J., 3 107) 198), pp [15] H. Triebel, Theory of function spaces, vol. 78 of Monographs in Mathematics, Birkhäuser Verlag, Basel, [16] H. Triebel, Theory of function spaces. III, vol. 100 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 006. Received: January, 01